What is a transcendental number?

Question

I came across some numbers which were called transcendental numbers. What are they exactly I want with explanation and eg

Answer 1

Transcendental numbers are those numbers which are not the root of a non-zero polynomial with integer coefficients.

The most famous example of a transcendental number is $\pi$

$\sqrt2$ is an irrational number but is not transcendental as it is the solution of the equation $x^2-2 = 0 $

You might be wondering how to prove $\pi$ is transcendental.

It can be proved by contradiction, suppose $\pi$ is algebraic, then $\pi i$ will also be algebraic as $i$ is algebraic and by Lindemann–Weierstrass theorem, we know that $e^{\pi i} = -1$ will be transcendental which is a contradiction as this is the solution of the equation $x^2 -1 = 0$

Answer 2

Trancedental numbers are numbers which are not the zero of a polynomial with coefficients in $\Bbb Q$. Actually there is a notion of transcendental numbers over any field, but it is usually referring to $\Bbb Q$ when the field is not specified.

For example, $\sqrt{2}$ is not transcendental, since it is a zero of the polynomial $x^2-2$. On the other hand, both $e$ and $\pi$ are transcendental, a difficult thing to prove.